Spin-1/2 square-root operator equation with Coulomb potential - a perturbation analysis

Seeking for a relativistic generalisation of the non-relativistic Schroedinger equation, one very soon arrives at equations with a square-root operator by having applied the quantum mechanical correspondence principle to the formula of relativistic energy. The problems of these equations are at least two fold: when coupled to an electromagnetic field, their relativistic invariance is not evident or even doubtful and due to their non-local character, it seems to be that they cannot be maintained mathematically in an easy way. For spin-1/2 particles, these difficulties can be overcome by the Dirac equation, which leads e.g. to the prediction of binding energies of an electron in a hydrogen atom that are compatible with experimental results up to the forth order of the fine structure constant, inclusively. Ignoring the problem with relativistic invariance, one may ask, if there exists a square-root equation, for which one can achieve the same good agreement with experiments for the latter physical system. It is going to be shown, that the answer to this question is affirmative and does not exceed the skills obtained in a course about non-relativistic quantum mechanics and physics of atoms, respectively.